Non-local Boxes. Philippe Lamontagne. July 31, 2014

Transcription

1 Non-local Boxes Philippe Lamontagne July 31,

2 Contents 1 Introduction Preliminaries The Non-signaling Polytope Deporalization Trivial Communication Complexity Two participants n participants Probabilistic Non-local Games Binary Games XOR-games Non-local Computation Non-locality Distillation Limits on Distillation Protocols Known Distillation Protocols Forster, Winkler, Wolf Protocol Brunner, Skrzypczyk Protocol Implications in Cryptography 16 7 Generalized Non-local Boxes Arbitrary Input or Output Size d-output Boxes d-input Boxes Interconversions of Non-local Correlations Multi-party Correlations Acknowledgments 22 2

3 1 Introduction The study of non-local boxes arouse from the study of quantum entanglement and from the question: why isn t entanglement more non-local?. Correlations stronger than quantum entanglement, but that still do not allow for instantaneous transmission of information have been known to exist [18]. 1.1 Preliminaries The concept of non-local boxes is inspired by that of quantum systems. They are closely related as a quantum system can be viewed as a non-local box, where the choice of measurement is the input and the outcome of the measurement is the output, and a non-local box can be viewed as a super-quantum system. Of course, not all boxes as defined under are non-local: they can be local, quantum, or super-quantum. Definition 1. A bipartite correlated box (or box) is a device with two ends, one of which is held by Alice, the other one by Bob. Each end has the following input-output behaviour: given input x on Alice s side (respectively y on Bob s side), the box will output a (respectively b) according to some probability distribution P (a, b x, y) where x, y, a, b {0, 1}. Throughout this paper, we will refer to boxes by their probability distributions. For convenience, we will also write P (ab xy) and P (a, b x, y) interchangeably. It is important to note that boxes are atemporal, meaning that the output comes out on one side as soon as an input is given. Was this not the case (if, for example, the box waits for both inputs before giving outputs), then one could transmit information to the other party by deliberately delaying it s input. Quantum entanglement does not allow for faster-than-light communication. This property is called non-signaling. Likewise, we are only interested in studying boxes that are non-signaling, which means Alice cannot learn anything from Bob s input by looking at her output. Definition 2. A box P is non-signaling if the sum over Bob s inputs of the joint probability distribution is equal to Alice s marginal distribution and vice 3

4 versa: P (a, b x, y) = b b P (a, b x, y) = a a It is signaling if it is not non-signaling. P (a, b x, y ) = P A (a x) a, x, y, y and P (a, b x, y) = P B (b y) b, x, x, y. Non-signaling correlations can be of many types. Of the boxes with this property, we find ones that can be implemented with classical theory, quantum theory or even super-quantum theory. Since the class of non-signaling correlations include quantum and classical ones, we may define a box as being non-local in the same way some quantum correlations are non-local. A box is said local if the output on one side depends only on the input on the same side. Local correlations can be simulated with only shared randomness by non-communicating participants. Definition 3. A box P is local if it can be written as P (a, b x, y) = i λ i Pi A (a x)pi B (b y) where λ i 0 and i λ i = 1. A box is non-local if it is not local. In essence, definition 3 says that any local box is a convex combination of local boxes. This is in accordance with the fact that the set of local correlations form a polytope[1] with the vertices being deterministic boxes (i.e. boxes with output uniquely determined). Now that we have defined what is non-locality, it would be useful to be able to quantify it. The value defined next is taken from the Clauser-Horne- Shimony-Holt inequality (or CHSH inequality)[9] which give an upper bound on local correlations. This inequality was designed as an application of Bell s famous theorem[3], but became a measure of non-locality. It was originally stated with expectation values of measurements of quantum system. We give a more information theoretic description from [14]. Definition 4. Let X xy (P ) = P (00 xy) + P (11 xy) P (01 xy) P (10 xy). The CHSH value of box P is CHSH(P ) = max X xy (P ) + X xȳ (P ) + X xy (P ) X xȳ (P ) xy 4

5 Clause, Horne, Shimony and Holt s derivation of Bell s theorem is stated in theorem 1 as a upper bound on the correlation of two local variables. It gives a necessary and sufficient condition on correlations for them to be local. Theorem 1 (Bell). A box P is local if and only if CHSH(P ) 2. Cirel son[8] later found an upper bound on the CHSH value that all quantum correlations must obey. It is a necessary condition for correlations to be achievable by quantum mechanics. Theorem 2 (Cirel son). If a box P can be implemented by quantum mechanics, then CHSH(P ) 2 2. However, this condition is not sufficient. This was remedied by [16] who found a necessary and sufficient condition on boxes for them to be quantum. Theorem 3. A box P can be implemented by a quantum state if and only if arcsin X xy + arcsin X xȳ + arcsin X xy arcsin X xȳ π. For any xy = 00, 01, 10, 11 where X xy is defined in definition 4. The following box was introduced by Popescu and Rohrlich [18] as a correlation achieving the maximal algebraic of 4 of the CHSH inequality. It is at the core of the study of non-locality and is used in the proofs of many of the results presented in this work. Definition 5. The Popescu-Rohrlich box (PR-box) is described by the following probability distribution: { P P R 1/2 if a b = xy (a, b x, y) = 0 otherwise. The noisy symmetric (or isometric) PR-boxes are the boxes of the form P ɛ = ɛp P R + (1 ɛ)p P R. where P P R is the anti-pr-box: P P R (ab xy) = 1/2 if a b xy. 5

6 2 The Non-signaling Polytope Barrett et al.[1] characterized the class of non-signaling correlations. All probability distributions within this class are subject to the following conditions: 1. positivity P (ab xy) 0; 2. normalization P (ab xy) = 1; 3. non-signaling constraints (see definition 2). a,b Since these constraints are linear, the class forms a polytope. To determine the dimension of the polytope, first note that the set of probabilities P (a, b x, y) where x, y, a, b {0, 1} form a table with 2 4 entries. The dimension of the polytope is then given by substracting the number of independant constraints from 2 4 which gives us the number of independant variables of the table, and turns out to be 8. The polytope has 24 vertices, 16 of which correspond to local deterministic boxes of the form { P αβγδ 1 if a = αx β, b = γy δ (a, b x, y) = 0 otherwise where α, β, γ, δ {0, 1}. These alone form the local polytope containing all local boxes as a convex combination of those 16 vertices. The remaining 8 vertices of the non-local polytope are of the form { P αβγ 1/2 if a b = xy αx βy γ (a, b x, y) = 0 otherwise where α, β, γ {0, 1}. Theorem 4. All vertices of the local polytope are equivalent under reversible local operations and all non-local vertices of the non-signaling polytope are equivalent under reversible local operations. 6

7 By reversible local operations, it is meant that Alice may relabel her input, x x 1, or she may relabel her output conditionaly on her input, a a αx β, and similarly for Bob. It is easy to see that any vertex of a given class (local or non-local) can be transformed into any other vertex of the same class by these operations. 2.1 Deporalization Deporalization is the act of taking a box which is a mixture of any nonsignaling box and transforming it into a symmetric box, while preserving the CHSH value. It consists of generating three maximally random bits α, β, γ and doing the following substitutions: x x α, y y β, a a βx αβ γ and b b αy γ. Note that this operation requires three bits of shared randomness between the two parties for every box they wish to depolarize. 3 Trivial Communication Complexity There has been evidence that non-locality helps in the communication complexity of some distributed tasks. See for example [4]. Protocols that make use of non-locality in the form of quantum entanglement offers advantages over local protocols, but since quantum non-locality is restricted, it is natural to ask ourselves if stronger non-locality is more helpful. Definition 6. The communication complexity of a function f is trivial if it can be computed using a single bit of communication per participant. This is the minimum communication needed to compute any function which is not itself trivial (i.e. it does not depend on only one of the inputs). 3.1 Two participants It will be useful to define the following property. Most proofs of trivial communication complexity using non-local boxes try to achieve this property. Definition 7. The boolean function f is distributively computed by Alice and Bob if they respectively receive x and y and output a and b such that a b = f(x, y). 7

8 Evidently, every function that can be distributively computed has trivial communication complexity. So every function that has communication complexity strictly greater than 1 cannot be distributively computed, and since most functions have non-trivial communication complexity, most functions are not distributively computable. Perhaps surprisingly, the next result by van Dam [20] shows that the existence of the NLB renders every function s communication complexity trivial. Theorem 5. In a world in which perfect non-local boxes exists, all boolean functions can be distributively computed. The proof uses the fact that every function f : {0, 1} n {0, 1} n {0, 1} can be expressed as a multi-variable polynomial which can be written in the form f(x, y) = i P i(x) Q i (y), where P i and Q i are polynomials and x, y {0, 1} n. This can be distributively computed by Alice and Bob because P i (x) depends only of x and Q i (y) only of y. They then input P i (x) and Q i (y) into the ith box. 3.2 n participants Let us extend the definition of distributed computation to n players, where the parity of the outputs is equal to the value of the function. Definition 8. The boolean function f is n-partite distributively computed by n participants if they respectively receive x i and output a i, 1 i n, such that n i=1 a i = f(x 1,..., x n ). The next result, by Barrett and Pironio [2], extends van Dam s result to n-partite communication complexity. Theorem 6. Correlations of the form { 1/2 n 1 if n i=1 P (a 1,..., a n x 1,..., x n ) = a i = f(x 1,..., x n ) mod 2 0 otherwise can be simulated with non-local boxes. Corollary 1. Any n-partite communication complexity problem can be solved with n 1 bits of communication. This is easy to see, as all participants send their outputs to the first who can then compute the function. 8

9 3.3 Probabilistic All triviality results presented thus far concern deterministic multipartite functions. Brassard et al. [5] found that this still applies when considering probabilistic multipartite computation. Theorem 7. In a world in which noisy non-local boxes which succeed more than % exist, all probabilistic functions can be distributively computed. 6 This lets us define the set of boxes that trivialize communication complexity. Corollary 2. Let B cc = 4 2/ , then all boxes P such that CHSH(P ) > B cc trivialize communication complexity. The CHSH value of a symmetric non-local box with probability of success is 4 2/3 and using the depolarisation protocol described in section 2.1, all boxes above CHSH value B cc trivialize communication complexity. 4 Non-local Games All the results of this section are due to Cleve et al.[10], except for the ones of section 4.3. Those last are from Linden et al. [17]. When playing a non-local game, Alice and Bob are space-like separated but allowed to share randomness. They are however allowed to elaborate a strategy beforehand. Alice and Bob respectively receive x X and y Y picked at random according to the probability distribution π. They must respectively output a A and b B. They win if V (a, b, x, y) = 1. Definition 9. A non-local game G = (X Y, A B, π, V ) consists of a set of inputs X Y, a set of outputs A B, a probability distribution π : X Y [0, 1] and a predicate V : X Y A B {0, 1}. Next is defined the best probability with which Alice and Bob can win a game when they are restricted to classical strategies, i.e., strategies that do not make use of non-locality. 9

10 Definition 10. The maximum winning probability for a classical strategy for a non-local game G = (X Y, A B, π, V ) is ω C (G) = max a,b π(x, y)v (a(x), b(y), x, y) x,y where the maximum is taken over all functions a : X A and b : Y B. If we allow Alice and Bob to share entanglement, their winning probability may benefit from it. A quantum strategy is determined by a bipartite state ψ shared between Alice and Bob. They both perform some measurement according to their respective input and output the result of that measurement. precisely, a quantum strategy consists of: a state ψ A B for A and B isomorphic copies of the vector space C n for some n. Where A represents Alice s part of ψ and B Bob s part; two sets of positive semidefinite n n matrices satisfying {X a x x X, a A} and {Y b y y Y, b B} a A X a x = I and b B y b y = I for every x X and y Y, where I is the n n identity matrix. We define the maximum winning probability of players with quantum strategies the following way. Definition 11. The maximum winning probability of a quantum strategy for a non-local game G = (X Y, A B, π, V ) is ω Q (G) = max ψ (x,y) X Y π(x, y) (a,b) A B ψ Xx a Yy b ψ V (a, b, x, y) 10

11 4.1 Binary Games In this section, we consider non-local games where answers are bits. Definition 12. A binary game G = (X Y, A B, π, V ) is a non-local game where A = B = {0, 1}. This next result states that quantum strategies cannot have an advantage over classical strategies if there exists a quantum strategy that always win the game. Theorem 8. Let G be a binary game. If there exists a quantum strategy for G that wins with probability 1, then ω C (G) = 1. This result is fairly strong, it implies that we will never be able to perfectly achieve such tasks when it is not possible classically. 4.2 XOR-games In this section, we study games for which the result depends not on the individual answers, but on the exclusive-or of respective answers. This category of games include the bipartite communication complexity tasks of section 3. Definition 13. A XOR-game G = (X Y, A B, π, V ) is a binary game where V : C X Y {0, 1} and C = {a b a A, b B}. The following definition will be of use for some of the results of this section. It is the winning probability when players are restricted to a trivial strategy, a trivial strategy consisting of outputting random bits. Definition 14. The success probability for a game G if both parties are restricted to a trivial strategy (output random bits) is τ(g) = 1 2 c {0,1} π(x, y)v (c, x, y) x,y When playing a XOR-game, the gain of the best quantum strategy over the trivial strategy cannot be too great compared to the gain of the best classical strategy over the trivial strategy. This is the essence of the following result, wich upper bounds the gap between quantum and classical advantages over the trivial strategy. 11

12 Theorem 9. Let G be a XOR-game. Then where K G is Grothendeick s constant. ω Q (G) τ(g) ω C (G) τ(g) K G Grothendeick s constant K G is the smallest number such that, for all integers N 2 and all N N real matrices M, if M(i, j)a i b j 1 for all numbers a 1,..., a N, b 1,..., b N in [ 1, 1], then M(i, j) u i v j K G i,j i,j for all unit vectors u 1,..., u N, v 1,..., v N in R n for any n. The exact value of Grothendieck s constant is not known, but it is known to satisfy π K G 2 log(1 + 2) Finally, the coming result upper bounds the maximum quantum winning probability by a function of the maximum classical winning probability. Theorem 10. Let G be a XOR-game. Then { γ 1 ω C (G) if ω C (G) γ 2 ω Q (G) sin 2 ( πω 2 C(G)) if ω C (G) > γ 2, where γ 1 and γ 2 are the solution to the equation π 2 sin(πγ 2) = sin2 ( π 2 γ 2) γ 2 = γ 1. γ and γ Non-local Computation Consider the scenario in which Alice and Bob wish to distributively compute a function whose input is also distributed. Alice and Bob respectively receive bit strings x and y and they must output single bits a and b such that a b = f(x y). What is particular in this type of non-local game is that neither of the players learn anything about the input since the individual bits of x and y are uniformly distributed from Alice and Bob s perspective. 12

13 Definition 15. A non-local computation game (or NLC-game) of a function f is a XOR-game G = (X Y, A B, π, V ) where V : C Z {0, 1}, Z = {x y x X, y Y } and V (a b, x y) = 1 if a b = f(x y). Linden et al. showed that when considering such a model, neither classical nor quantum strategies can always win a given game. Theorem 11. Let G be a non-local computation game. Then ω C (G) = ω Q (G) < 1 5 Non-locality Distillation The motivation behind the study of non-locality distillation is the question of whether we can use a set of boxes to simulate the behaviour of a more non-local one. For example, can we use a set of n noisy PR-boxes to simulate the behaviour of a less noisy PR-box. Definition 16. A non-locality distillation protocol (NDP) consists of local operations performed by Alice and Bob on their respective ends of n boxes with a given CHSH value to simulate the input-output behavior of a higher valued box. A non-locality distillation protocol N on n boxes P, denoted N n [P ], consists of local operations performed on the boxes to simulate the input/output behaviour of a box P = N n [P ]. Of course, for a distillation protocol to be useful we must have that the CHSH value of the box simulated by the protocol is greater than the CHSH value of the input boxes (i.e. CHSH(P ) > CHSH(P )). However, we do not require that distillation is achieved for all families of boxes, because as we will soon see, this would be impossible. 5.1 Limits on Distillation Protocols This section contains upper bounds and impossibilities on non-locality distillation protocol. For example, the first statement of theorem 12 asserts that no distillation protocol can create non-locality from locality. The results of this theorem were taken from [12]. Theorem 12. For any non-locality distillation protocol N, 13

14 if CHSH(P ) 2 then CHSH(N [P ]) 2; if P is a box whose correlations are achievable by quantum mechanics, then CHSH(N [P ]) 2 2; if CHSH(P ) < 4 then CHSH(N [P ]) < 4. It is important to understand that the second statement of theorem 12 applies only to correlations that can be obtained by measurements on quantum states. As we will see in section 5.2.1, some protocols bring boxes of CHSH value near 2 and brings them to 3 > 2 2, but these cannot be simulated by measurements on quantum states. Short[19] proved the impossibility of distillation protocols operating on two copies of noisy PR-boxes. Theorem 13. Two copies of a noisy PR-box cannot be distilled. For any P ɛ = ɛp P R + (1 ɛ)p P R, there is no N 2 such that CHSH(N 2 [P ɛ ]) > CHSH(P ɛ ). His proof, which applies to more general frameworks than just non-local boxes, works by showing that the probability that the protocol simulates a PR-box as a function of the same probability for the initial boxes is a polynomial of degree two in the original probability. He then shows a set of constraints that no polynomial of degree two can satisfy. 5.2 Known Distillation Protocols In this section, we consider only protocols achieving distilliation of correlations outside the quantum set. All known distillation protocols are applied to the same family of boxes, termed correlated non-local boxes by Brunner and Skrzypczyk[6]. Correlated non-local boxes are of the form Pɛ C = ɛp P R + (1 ɛ)p C where P C is the fully correlated box P C (ab xy) = 1/2 if a b = 0. Correlated non-local boxes have a CHSH value of 2(ɛ + 1) > 2. What characterizes these boxes is their bias towards correlated outputs, i.e. a b = 0. This means that when the box outputs uncorrelated bits, you are assured that it has output the correct answer. Both protocols presented here will make use of this fact. Non-locality distillation protocols can however be applied to any box of the non-signaling polytope. Whether a protocol distills or not a given box depends on its joint probability distribution. 14

15 5.2.1 Forster, Winkler, Wolf Protocol The first non-locality distillation protocol for non-local boxes was discovered by Forster, Winkler and Wolf (FWW)[14]. Their protocol is fairly simple, it uses the parity of the output of the initial boxes as output. F n [P C ɛ ](x, y) 1. On inputs x and y, input x and y into all n boxes; 2. Let a i and b i be the outputs of the ith box, output a = n i=1 a i and b = n i=1 b i. This protocol achieves distillation. Theorem 14. For n > 1 and 0 < ɛ < 1/2, CHSH(F n [P C ɛ ]) = 3 (1 2ɛ) n > 3 (1 2ɛ) = CHSH(P C ɛ ). Perhaps interestingly, Peter Hoyer and Jibran Rashid showed in unreleased work that when restricted to input x and y into all boxes, the FWW protocol is optimal Brunner, Skrzypczyk Protocol This protocol, introduced in [6], operates on two boxes. Unlike the FWW protocol which brings correlated value to a CHSH value of 3 in the asymptotic limit, the Brunner Skrzypczyk protocol brings then to the CHSH value of 4 in the asymptotic limit. Which means they cross the communication complexity bound B cc, increasing the class of correlations that trivialize communication complexity. B 2 [P C ɛ ](x, y) 1. Input x, y into first box; 2. Let a 1 and b 1 be the outputs of the first box, input x a 1 and y b 1 into second box; 3. Let a 2 and b 2 be the outputs of the second box, output a = a 1 a 2 and b = b 1 b 2. Theorem 15. For 0 < ɛ < 1, CHSH(B 2 [P C ɛ ]) = 3ɛ ɛ > 2(ɛ + 1) = CHSH(P C ɛ ). 15

16 When applied to boxes of the form ɛp P R + δp P R + (1 ɛ δ)p C for 0 < δ < ɛ < 1, which are achievable by quantum states, the protocol still achieves distillation for some values of ɛ and δ (without crossing tsirelson s bound of course). Corollary 3. There exists correlations arbitrarily close the the classical and quantum sets of correlations that trivialize communication complexity. The Brunner Skrzypczyk protocol brings boxes of CHSH value arbitrarily close to 2, yet still unreachable by quantum states, and distills then to CHSH value arbitrarily close to 4 crossing the bound B cc defined in corollary 2 6 Implications in Cryptography Definition 17. An oblivious transfer (OT) protocol is a protocol in which a sender sends a message to the receiver with probability 1/2, while himself learning nothing of whether the receiver received the message. One out of two oblivious transfer (1-2 OT) is a variant in which the sender holds two bits s 0 and s 1, and the receiver has bit c. The receiver wishes to learn bit s c without the sender learning c. Wolf and Wullschleger [21] gave a protocol for secure 1-2 OT. Their protocol uses a single PR-box and procedes as follow. Alice inputs x = x 0 x 1. Bob inputs y = c. Alice gets output a and Bob b. Alice sends m = x 0 a to Bob. Bob computes m b = x 0 a b = x 0 (x 0 x 1 )c = x c. Wolf and Wullschleger s protocol for 1-2 OT is secure, but when trying the usual reduction from OT to 1-2 OT, it becomes insecure. In the reduction, the sender uses s k = b and s k = 0 with k R {0, 1} the receiver uses any c {0, 1}. The players perform 1-2 OT with s k, s k and c, then the sender announces k to the receiver who learns b with probability 1/2 if k = c. Using their protocol, the receiver can delay his input into the box until the sender announced k and always learn b. Buhrman et al. [7], based on the Wolf and Wullschleger protocol, showed that bit-commitment and OT are possible given perfect PR-boxes. The following definition will be of use in the bit-commitment protocol described in Buhrman et al. Definition 18. Let the operator x 11 for a bit string x denote the number of substrings 11 of x starting at odd positions (with positions starting at 1). 11 is defined recursively as follows 16

17 ɛ 11 = 0 where ɛ is the empty string; abx 11 = x if ab = 11, x 11 otherwise. Burhman et al. s protocol for bit-commitment consists of repeating k times the following commit/reveal scheme: Bit-Commitment(c) Commit Reveal Alice wants to commit to bit c. She constructs x {0, 1} 2n+1 by randomly choosing the first 2n bits and choosing the last bit such that x 1... x 2n + x 2n+1 + c is even. Alice inputs the bits x 1,... x 2n+1 into the 2n + 1 PR-boxes. Let a 1,..., a 2n+1 be the outputs. Alice computes A = i a i and sends it to Bob. Bob chooses a random string y R {0, 1} 2n+1 and inputs bits y 1,..., y 2n+1 into his end of the 2n+1 PR-boxes. Let b 1,..., b 2n+1 be the outputs. Alice sends c, x and b 1,..., b 2n+1 to Bob. Bob checks if a i b i = x i y i for 1 i 2n + 1 and x 1... x 2n 11 + x 2n+1 + c is even. If not, he accuses Alice of cheating. Theorem 16. This protocol is secure against Alice. The best probability with which Alice can change her mind is 1/2 + 1/2 k 1. This protocol is secure against Bob. The best probability with which Bob can learn c before the reveal stage is 1/2 + k/2 n+1. 7 Generalized Non-local Boxes In this section we study a more general class of non-local boxes, where we extend the set of inputs, the set of outputs, and the number of participants. We also present some results on the connections between types of generalized boxes. 17

18 7.1 Arbitrary Input or Output Size Consider boxes with binary inputs, but with outputs taken from arbitrary finite sets. Let d x denote the number of inputs and d a the number of outputs on Alice s side, similarily d y denotes the number of inputs and d b the number of outputs on Bob s side. Such boxes correspond to definition 1 but where x {0,..., d x 1}, y {0,..., d y 1}, a {0,..., d a 1} and b {0,..., d b 1}. We will refer to these as generalized boxes d-output Boxes The class of generalized boxes with d x = d y = 2 form a polytope P described by Barrett et al. [1]. It s dimension is 4d a d b 2d a 2d b. So if d a = d b = d, the dimension is 4d 2 4d and when d = 2 we find the dimension of the non-signaling polytope of section 2. They also found that the non-local vertices of this polytope are all equivalent under reversible local operations. A result analogous to the fact that all non-local vertices of the two-input two-output polytope are equivalent to the PR-box. Theorem 17. Every non-local vertice of P is equivalent under reversible local operations to { 1/k if (b a) xy mod k P (a, b x, y) = 0 otherwise. for some k {2,..., min{d a, d b }} where x, y {0, 1} and a, b {0,..., k 1}. Actually, for every k, the box described above is a representant of an equivalence class of non-local vertices. When d a = d b = k = d, this box violates the d-dimensional generalization of the CHSH inequality [11] up to it s algebraic maximum. We will refer to such boxes as d-output boxes d-input Boxes Jones and Masanes [15] characterized the set of generalized boxes for d a = d b = 2 and arbitrary d x and d y. Every class of non-local vertices for a given d x and d y is represented by a box parameterized by two integers g x {2,..., d x } 18

19 and g y {2,..., d y }. The box s behavious is non-deterministic for g x of the inputs and deterministic for d x g x of the inputs on Alice s side and analogously for Bob. We send the reader to their paper for the detailed description of the box. 7.2 Interconversions of Non-local Correlations This section covers the relations existing between different types of generalized non-local boxes. Theorems 18 and 19 are from [1]. Theorem 18. The following interconversions are possible: 1 d-output box and 1 d -output box can simlate 1 dd -output box 1 dd -output box can simulate 1 d-output box n d-output boxes can approximate 1 d -output box Lemma 1. Using n d-output boxes, Alice and Bob can exactly simulate at most n d -output boxes, for d d. Lemma 2. Using n d -output boxes, Alice and Bob can exactly simulate at most n(1 + log 2 d )/(1 + log 2 d) < n d-output boxes for d d. Theorem 19. It is in general impossible, using local reversible operations, to exactly simulate m d -output boxes from n d-output boxes. Theorem 19 follows from lemmas 1 and 2. It implies that there is little interconvertibility between families of d-output boxes. The following results by Dupuis et al [13] furthers this lack of interconversions by providing impossibilities of interconversions for d-boxes. This first theorem states that any finite amount of PR-boxes cannot exactly simulate a single 3-box. Theorem 20. It is impossible to simulate a 3-box exactly using a finite number of 2-boxes, infinite shared randomness and no communication. Their next theorem generalizes their first one. Theorem 21. Let S be a finite set of generalized non-local box with d x = d y = 2 and arbitrary d a and d b. Then there exists p such that the p-box cannot be simulated by a finite number of boxes taken from the set S. 19

20 The following results by Jones and Masanes [15] show that, to the contrary, d-input boxes are very interconvertible. Theorem 22. PR-boxes are sufficient to simulate all non-signaling correlations with binary output (d a =d b =2). Theorem 23. All correlations with arbitrary d x and d y, and binary output (d a = d b =2) are interconvertible. We refer the reader to the original paper for both proofs. 7.3 Multi-party Correlations Now consider the case where three participants Alice, Bob and Charlie exhibit non-local correlations. The definition of the non-local box can be extented to accomodate this new model. Definition 19. A tripartite correlated box (or box) is a device with three ends. Each end has the following input-output behaviour: given input x, y and z, the box will respectively output a, b and c according to some probability distribution P (abc xyz) where x, y, z, a, b, c {0, 1}. The probabilities P (abc xyz) are subject to positivity and normalization, and the trivial extension of the non-signaling constraints P (abc xyz) = P (abc x yz) b, c, x, x, y, z a a P (abc xyz) = P (abc xy z) b, c, x, y, y, z b b P (abc xyz) = P (abc xyz ) b, c, x, y, z, z c c While the non-signaling condition is roughly unchanged, non-locality needs to be defined differently than with bipartite correlations. Alice, Bob and Charlie can be pairwise local wich each other. Definition 20. A box P fully local if it can be written as P (abc xyz) = i λ i Pi A (a x)pi B (b y)pi C (c z) where λ i 0 and i λ i = 1. 20

21 Can also occur the situation where Alice and Bob are non-local, but they are local versus Charlie. We call such boxes two-way local along with any box which is a convex combination of such boxes. Definition 21. A box P is two-way local if it can be written as P (abc xyz) = p 1 λ i Pi AB (ab xy)pi C (c z) i + p 2 i + p 3 i λ i P AC i (ac xz)pi B (b y) λ i Pi BC (bc yz)pi A (a x) where the p i s and λ i s are positive and normalized. The set of non-local tripartite correlations form a 26 dimensions polytope.the set of local correlations form a subpolytope of the two-way local polytope, itself a subpolytope of the non-signaling polytope. Vertices of the local polytope correspond to boxes for which all outputs are deterministic, they are equivalent under reversible local operations to { 1 if a = 0, b = 0, c = 0 P (abc xyz) = 0 otherwise. Two-way local vertices are boxes that describe a PR-box shared between two players while the third has a deterministic box. They are equivalent under reversible local operations to { 1/2 if a b = xy and c = 0 P (abc xyz) = 0 otherwise. Non-local vertices are more complex than the two other types. The set of non-local vertices has 44 different classes of vertices, which we won t enumerate. One of these class is equivalent under reversible local operations to the natural extension of the non-local box { 1/4 if a b c = xyz P (abc xyz) = 0 otherwise. As with the generalized bipartite non-local boxes, it is possible to perform conversions between tripartite boxes. One could also be interested in the simulation of tripartite boxes using PR-boxes.We will, however, not go into further details about these. 21

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